Pacific Journal of Mathematics

EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS

JULIA WEBER

Volume 231 No. 1 May 2007

PACIFIC JOURNAL OF MATHEMATICS Vol. 231, No. 1, 2007

EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS

JULIA WEBER

For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f : X → X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic to a fixed point free G-map if the gener- alized equivariant Lefschetz invariant λG( f ) is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem.

1. Introduction

The Lefschetz number is a classical invariant in algebraic topology. If the Lefschetz number L( f ) of an endomorphism f : X → X of a compact CW-complex is nonzero, then f has a fixed point. This is the famous Lefschetz fixed point theorem. The converse does not hold: If the Lefschetz number of f is zero, we cannot conclude f to be fixed point free. A more refined invariant which allows to state the converse is the Nielsen num- ber: The Nielsen number N( f ) is zero if and only if f is homotopic to a fixed point free map. More generally, the Nielsen number is used to give precise minimal bounds for the number of fixed points of maps homotopic to f . Its development was started by Nielsen [1920]; a comprehensive treatment can be found in [Jiang 1983]. We are interested in the equivariant generalization of these results. Given a discrete group G and a G-equivariant endomorphism f : X → X of a finite proper G-CW-complex, we introduce equivariant Nielsen invariants called NG( f ) and N G( f ). They are equivariant analogs of the Nielsen number and are derived from the generalized equivariant Lefschetz invariant λG( f ) [Weber 2005, Defi- nition 5.13].

MSC2000: primary 55M20, 57R91; secondary 54H25, 57S99. Keywords: Nielsen number, discrete groups, equivariant, Lefschetz fixed point theorem.

239 240 JULIA WEBER

We proceed to show that these Nielsen invariants give minimal bounds for the number of orbits of fixed points in the G-homotopy class of f . One even obtains results concerning the type and “location” (connected component of the relevant fixed point set) of these fixed point orbits. These lower bounds are sharp if X is a cocompact proper smooth G-manifold satisfying the standard gap hypotheses. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem: If X is a G-Jiang space as defined in Definition 5.2, then LG( f ) = 0 implies that f is G-homotopic to a fixed point free map. Here, LG( f ) is the equivariant Lefschetz class [Luck¨ and Rosenberg 2003a, Definition 3.6], the equivariant analog of the Lefschetz number. These results were motivated by work of Luck¨ and Rosenberg [2003a; 2003b]. For G a discrete group and an endomorphism f of a cocompact proper smooth G-manifold M, they prove an equivariant Lefschetz fixed point theorem [2003a, Theorem 0.2]. The converse of that theorem is proven here. Another motivation for the present article is the fact that the algebraic approach to the equivariant Reidemeister trace provides a good framework for computation. The connection to the machinery used in the study of transformation groups [Luck¨ 1989; tom Dieck 1987] allows results to translate more readily from transformation groups to geometric equivariant topology and vice-versa. When G is a compact Lie group, Wong [1993] obtains results on equivariant Nielsen numbers which strongly influenced us. The main difference between our work and Wong’s is that we treat possibly infinite discrete groups. Another dif- ference is that our approach is more structural. We can read off the equivariant Nielsen invariants from the generalized equivariant Lefschetz invariant λG( f ). In case G is a finite group, Ferrario [2003] studies a collection of generalized Lefschetz numbers which can be thought of as an equivariant generalized Lefschetz number. In contrast to the generalized equivariant Lefschetz invariant λG( f ) these do not incorporate the WH-action on the fixed point set X H . A generalized Lef- schetz trace for equivariant maps has also been defined by Wong, as mentioned in [Hart 1999]. For compact Lie groups, earlier definitions of generalized Lefschetz numbers for equivariant maps were made in [Wilczynski´ 1984; Fadell and Wong 1988]. These authors used the collection of generalized Lefschetz numbers of the maps f H , for H < G. In general, these numbers are not sufficient since they do not take the equivariance into account adequately. For further reading on equivariant fixed point theory, see [Ferrario 2005], where an extensive list of references is given.

Organization of paper. In Section 2, we introduce the generalized equivariant Lef- schetz invariant. We briefly assemble the concepts and definitions which are needed for the definition of equivariant Nielsen invariants in Section 3. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 241

The equivariant Nielsen invariants give lower bounds for the number of fixed point orbits of f . This is shown in Section 4. The standard gap hypotheses are introduced, and it is shown that under these hypotheses these lower bounds are sharp. In the nonequivariant case, we know that the generalized Lefschetz invariant is the right element when looking for a precise count of fixed points. We read off the Nielsen number from this invariant. In general, the Lefschetz number contains too little information. But under certain conditions, we can conclude facts about the Nielsen number from the Lefschetz number directly. These are called the Jiang conditions [1983, Definition II.4.1] (see also [Brown 1971, Chapter VII]). In Section 5, we introduce the equivariant version of these conditions. We also give examples of G-Jiang spaces. In Section 6, we derive equivariant analogs of statements about Nielsen numbers found in [Jiang 1983], generalizing results from [Wong 1993] to infinite discrete groups. In particular, if X is a G-Jiang space, the converse of the equivariant Lefschetz fixed point theorem holds.

2. The generalized equivariant Lefschetz invariant

Classically, the Nielsen number is defined geometrically by counting essential fixed point classes [Brown 1971, Chapter VI; Jiang 1983, Definition I.4.1]. Alternatively one defines it using the generalized Lefschetz invariant. Let X be a finite CW-complex, let f : X → X be an endomorphism, let x be a basepoint of X, and let X λ( f ) = nα · α ∈ ޚπ1(X, x)φ

α∈π1(X,x)φ be the generalized Lefschetz invariant associated to f [Reidemeister 1936; Wecken 1941], where

−1 ޚπ1(X, x)φ := ޚπ1(X, x)/φ(γ )αγ ∼ α, with γ, α ∈ π1(X, x).

Here φ is the map induced by f on the fundamental group π1(X, x). We have φ(γ ) = wf (γ )w−1, where w is a path from x to f (x). The generalized Lefschetz invariant is also called Reidemeister trace in the literature, which goes back to the original name “Reidemeistersche Spureninvariante” used by Wecken. The set π1(X, x)φ is often denoted by ᏾( f ) and called set of Reidemeister classes of f . Since we will introduce a variation of this set in Definition 2.3, we prefer to stick with the notation used in [Weber 2005]. 242 JULIA WEBER

Definition 2.1. The Nielsen number of f is defined by  N( f ) := # α nα 6= 0 .

The Nielsen number is the number of classes in π1(X, x)φ with nonzero coeffi- cients. A class α with nonzero coefficient corresponds to an essential fixed point class in the geometric sense. In the equivariant setting, the fundamental category replaces the fundamental group. The fundamental category of a topological space X with an action of a discrete group G is defined as follows [Luck¨ 1989, Definition 8.15]. Definition 2.2. Let G be a discrete group, and let X be a G-space. Then the fundamental category 5(G, X) is the following category: • The objects Ob5(G, X)
are G-maps x : G/H → X, where the H ≤ G are subgroups. • The morphisms Morx(H), y(K )
are pairs (σ, [w]), where – σ is a G-map σ : G/H → G/K – [w] is a homotopy class of G-maps w : G/H × I → X relative G/H ×∂ I such that w1 = x and w0 = y ◦ σ . The fundamental category is a combination of the orbit category of G and the fundamental groupoid of X. If X is a point, then the fundamental category is just the orbit category of G, whereas when G is the trivial group, the definition reduces to the definition of the fundamental groupoid of X. We often view x as the point x(1H) in the fixed point set X H . We call X H (x) the connected component of X H containing x(1H). We also consider the relative H >H
>H H fixed point set, the pair X (x), X (x) . Here X (x) = {z ∈ X (x) | Gz 6= H} is the singular set, where Gz denotes the isotropy group of z. In order to H simplify notation, we use f (x) to denote f |X H (x), and we use fH (x) instead of f | . X H (x),X >H (x)
Fixed points of f can only exist in X H (x) when X H ( f (x)) = X H (x), i.e, when the points f (x) and x lie in the same connected component of X H . Definition 2.3. For x ∈ Ob 5(G, X) with X H ( f (x)) = X H (x) and a morphism v = (id, [w]) ∈ Mor( f (x), x), set H
:= H
−1 ∼ ޚπ1 X (x), x φ0 ޚπ1 X (x), x /φ(γ )αγ α, H −1 where α ∈ π1(X (x), x), γ ∈ Aut(x) and φ(γ ) = vφ(γ )v ∈ Aut(x). The automorphism group Aut(x) of the object x in the category 5(G, X) is a group extension lying in the short exact sequence H 1 → π1(X (x), x) → Aut(x) → WHx → 1. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 243

H Here WH := NG H/H is the Weyl group of H, it acts on X . We call WHx the subgroup of WH which fixes the connected component X H (x). Groups obtained from different choices of the path w and of the point x in its isomorphism class x¯ are canonically isomorphic, so those choices do not play a H
role. The group ޚπ1 X (x), x φ0 generalizes the group ޚπ1(X, x)φ defined above. So it can be seen as the free abelian group generated by equivariant Reidemeister H H classes of f (x) with respect to the action of WHx on X (x). A map (σ, [w]) ∈ Mor(x, y) induces a group homomorphism [ ] ∗ : K
→ H
(σ, w ) ޚπ1 X (y), y φ0 ޚπ1 X (x), x φ0 by twisted conjugation, and we know that the induced group homomorphism is the same for every map in Mor(x, y) [Weber 2005, Lemma 5.2]. The generalized equivariant Lefschetz invariant [Weber 2005, Definition 5.13], λG( f ), is an element in the group

M H 3G(X, f ) := ޚπ1(X (x), x)φ0 . x¯∈Is 5(G,X), X H ( f (x))=X H (x) Here Is 5(G, X) denotes the set of isomorphism classes of the category 5(G, X). Geometrically, it corresponds to the set of WH-orbits of connected components X H (x) of the fixed point sets X H , for (H) ∈ consub(G), i.e., for a set of represen- ' tatives of conjugacy classes of subgroups of G. There is a bijection Is 5(G, X) −→ H q(H)∈consub(G)WH \ π0(X ) which sends x : G/H → X to the orbit under the H H H WH-action on π0(X ) of the component X (x) of X which contains the point x(1H) [Luck¨ and Rosenberg 2003a, Equation 3.3]. Let ^f H (x) and f^>H (x) denote the lift of f H (x) to the universal covering space X^H (x) and to the subset X^>H (x) ⊆ X^H (x) that projects to X >H (x) under the covering map. At the summand indexed by x¯, the generalized equivariant Lefschetz invariant is given by

ޚ Aut(x) H >H
H λG( f )x¯ := L ^f (x), f^(x) ∈ ޚπ1(X (x), x)φ0 , where the refined equivariant Lefschetz number [Weber 2005, Definition 5.7] ap- pears on the right hand side. It is defined by

ޚ Aut(x) H >H
X p c H >H L ^f (x), f^(x) := (−1) trޚ Aut(x)(C p( ^f (x), f^(x))), p≥0 where the trace map trޚ Aut(x) [Weber 2005, Definition 5.4] is induced by the pro- H P P jection ޚ Aut(x) → ޚπ (X (x), x) 0 , r · g 7→ H r ·g ¯. 1 φ g∈Aut(x) g g∈π1(X (x),x) g Instead of ޚ, other rings can be used. 244 JULIA WEBER

This trace map generalizes the trace map used in [Luck¨ and Rosenberg 2003a], and the refined equivariant Lefschetz number is a generalization of the orbifold Lefschetz number (Definition 1.4 of that reference). The refined equivariant Lefschetz number Lޑ Aut(x) ^f H (x)
will be particularly important to us, so we give some formulas describing it. For a finite proper G- CW-complex X we have [Weber 2005, Lemma 5.9]

ޑ Aut(x) H
X p X −1 H L ^f (x) = (−1) |Ge| · incφ( f, e) ∈ ޑπ1(X (x), x)φ0 .

p≥0 G·e∈G\Ip(X)

Here I p(X) denotes the set of p-cells of X, e runs through the equivariant cells of X, and Ge is its isotropy group. The refined incidence number [Weber 2005, H Definition 5.8] incφ( f, e) ∈ ޚπ1(X (x), x)φ0 for a p-cell e ∈ I p(X) is defined to be the “degree” of the composition

−1 ie _ 0 0 h∼ f h ∼ _ 0 0 e¯/∂e −→ e /∂e −→ X p/ X p−1 −→ X p/ X p−1 −−−→ e /∂e 0 0 e ∈Ip(X) e ∈Ip(X)

prπ·¯e/∂e · −−−−→ π ·e ¯/∂e −→ πφ0 ·e ¯/∂e.

Here e¯ is the closure of the open p-cell e and ∂e =e ¯\e. The map ie is the inclusion, h is a homeomorphism and prπ·¯e/∂e is the projection. If X = M is a cocompact proper G-manifold, we have [Weber 2005, Theo- rem 6.6]

ޑ Aut(x)
X −1 H
c
^H = H − · L f (x) (WHx )z deg idTz M (x) Tz( f (x)) αz. WHx ·z∈ WHx \Fix( f H (x)) Here the map on the tangent space is extended to the one-point compactification H
c Tz M (x) . The relative versions of these formulas also hold. ޑ Aut(x) ^H
=
We have L f (x) chG(X, f ) λG( f ) x¯ [Weber 2005, Lemma 6.4], L K where chG(X, f ) : 3G(X, f ) → y¯∈Is 5(G,X) ޑπ1(X (y), y)φ0 is the character ޑ Aut(x) H
map [Weber 2005, Definition 6.2]. So we can derive L ^f (x) from λG( f ). The equivariant analog of the Lefschetz number is the equivariant Lefschetz L class LG( f ) ∈ x¯∈Is 5(G,X),X H ( f (x))=X H (x) ޚ, whose value at x¯ is

ޚWHx H >H
LG( f )x¯ = L f (x), f (x) H [Luck¨ and Rosenberg 2003a, Definition 3.6]. The projection of π1(X (x), x)φ0 to the trivial group {1} induces an augmentation map sending λG( f ) to LG( f ): M H M s : ޚπ1(X (x), x)φ0 → ޚ. x¯∈Is 5(G,X), x¯∈Is 5(G,X), X H ( f (x))=X H (x) X H ( f (x))=X H (x) EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 245

3. Equivariant Nielsen invariants

P H H Given an element α nα ·α ∈ ޚπ1(X (x), x)φ0 , we call a class α ∈ π1(X (x), x)φ0 essential if the coefficient nα is nonzero. Let G be a discrete group and let X be a cocompact proper smooth G-manifold. Let f : X → X be a smooth G-equivariant map such that Fix( f )∩∂ X = ∅ and such ∈ − that for every z Fix( f ) the determinant of the map (idTz X Tz f ) is different from zero. One can always find a representative in the G-homotopy class of f which satisfies this assumption. Since the generalized equivariant Lefschetz invariant is G-homotopy invariant, we can replace f by this representative if necessary. Definition 3.1. The equivariant Nielsen class of f is −
X det idTz X Tz( f ) νG( f ) = · αz. detid −T ( f )
Gz∈G\Fix( f ) Tz X z

G −1 Here αz ∈ π1(X z (z), x) is the loop given by [t ∗ f (t) ∗ w], where x is a basepoint in X Gz (z), t is a path from x to z and w is a path from f (x) to x. The basepoint x may differ from z, e.g., if we have more than one fixed point in a connected component of X Gz . If x = z, we may choose t and w to be constant. The equivalence relation assures that this definition is independent of the choices involved. We can also derive the equivariant Nielsen class νG( f ) from the generalized equivariant Lefschetz invariant λG( f ). Lemma 3.2. The invariant ν( f ) is the image of the generalized equivariant Lef- schetz invariant λ( f ) under the quotient map where we divide out the images of nonisomorphisms:

M H ∗ νG( f ) = λG( f ) ∈ ޚπ1(X (x), x)φ0 /{Im(σ, [w]) | σ nonisom.} x¯∈Is 5(G,X), X H ( f (x))=X H (x) Proof. We consider the equation obtained in the refined equivariant Lefschetz fixed point theorem [Weber 2005, Theorem 0.2]. We have

= X ◦ Gz − c λG( f ) 3G(z, f ) indGz ⊆G(Deg0 ((idTz X Tz f ) )). Gz∈G\Fix( f )

Gz Here Deg0 is the equivariant degree [Luck¨ and Rosenberg 2003a], it has values in the Burnside ring A(Gz). On basis elements [Gz/L] ∈ A(Gz), the map 3G(z, f )◦ indGz ⊆G is given by ◦ [ ] = [ ] ∗ 3G(z, f ) indGz ⊆G( Gz/L ) (pr, cst ) αz, 246 JULIA WEBER where cst denotes the constant map and

∗ Gz L (pr, [cst]) : ޚπ1(X (z), x)φ0 → ޚπ1(X (z ◦ pr), x ◦ pr)φ0 is the map induced by the projection pr : Gz/L → Gz/Gz. Gz − c We know that Deg0 ((idTz X Tz f ) ) is a unit of the Burnside ring A(Gz) since Gz − c
2 = Deg0 ((idTz X Tz f ) ) 1 [Luck¨ and Rosenberg 2003a, Example 4.7]. In gen- eral a unit of the Burnside ring A(Gz) may consist of more than one summand [tom Dieck 1979]. The summand [Gz/Gz] is always included with a coefficient +1 or −1, but there might be summands [Gz/L] for L < Gz appearing. So one fixed point might give more than one class with nonzero coefficients. If we divide out the images of nonisomorphisms, then we divide out the im- ∗ age of (pr, [cst]) for all L 6= Gz. We are left with the summand ±αz coming from ±1[Gz/Gz]. This cannot lie in the image of any nonisomorphism. So each fixed point leads to exactly one summand. The sign is the sign of the determinant −
det idTz X Tz( f ) , so the claim follows.  We set

H H ∗ ޚπ1(X (x), x)φ00 := ޚπ1(X (x), x)φ0 /{Im(σ, [w]) | σ nonisom.}.

We use the equation established in Lemma 3.2 to define νG( f ) directly for all endomorphisms of finite proper G-CW-complexes.

Definition 3.3. Let X be a finite proper G-CW-complex, and let f : X → X be an equivariant endomorphism. Then the equivariant Nielsen class of f is

M H νG( f ) := λG( f ) ∈ ޚπ1(X (x), x)φ00 . x¯∈Is 5(G,X), X H ( f (x))=X H (x)

We define equivariant Nielsen invariants by counting the essential classes α of H ޑ Aut(x) H
H νG( f )x¯ in ޚπ1(X (x), x)φ00 and of L ^f (x) in ޑπ1(X (x), x)φ0 . Definition 3.4. Let G be a discrete group, let X be a finite proper G-CW-complex, and let f : X → X be a G-equivariant map. Then the equivariant Nielsen invariants of f are elements

G M NG( f ), N ( f ) ∈ ޚ x¯∈Is 5(G,X) defined for x¯ with X H ( f (x)) = X H (x) by  NG( f )x¯ := # essential classes of νG( f )x¯ , EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 247

G n [ K := ⊆ 0 ¯ ≥ ¯ N ( f )x¯ min #Ꮿ Ꮿ π1(X (y), y)φ such that for all z x and y≥x ޑ Aut(z)
for all essential classes α of L f^Gz (z) there are ∗ o β ∈ Ꮿ and (σ, [t]) ∈ Mor(z, yβ ) such that (σ, [t]) (β) = α .

We continue them by 0 to x¯ ∈ Is 5(G, X) with X H ( f (x)) 6= X H (x). G H Note that NG( f )x¯ = NG( fH (x)) and N ( f )x¯ = NOG( f (x)) in the notation of [Wong 1993]. Thus the invariants defined here using the algebraic approach are equivalent to the invariants defined using the classical covering space approach of Wong. An essential class α of Lޑ Aut(x) ^f H (x)
corresponds to an essential fixed point H class of f (x), a WHx -orbit of fixed points which one cannot get rid of under any G-homotopy, as can be seen from the refined orbifold Lefschetz fixed point theorem [Weber 2005, Theorem 6.6]. An essential class α of νG( f )x¯ corresponds H to an essential fixed point class of fH (x), an orbit of fixed points on X (x) \ X >H (x) that cannot be moved into X >H (x). Counting the essential classes will give us information on the number of fixed points and fixed point orbits. The equivariant Nielsen invariants are G-homotopy invariant since they are de- rived from λG( f ), which is itself G-homotopy invariant. 0 Proposition 3.5. Given a G-homotopy f 'G f , we have

0 G G 0 NG( f ) = NG( f ), N ( f ) = N ( f ).

0 0 Proof. If f 'G f , with a homotopy H : X × I → X such that H0 = f and H1 = f , then by invariance under homotopy equivalence [Weber 2005, Theorem 5.14] we −1 ∼ 0 have an isomorphism 3G(i1) 3G(i0) : 3G(X, f ) −→ 3G(X, f ) which sends 0 λG( f ) to λG( f ). The isomorphisms 3G(i1) and 3G(i0) are given by composition of maps, so they do not change the number of essential classes. They also do not change the property of a class to lie in the image of a nonisomorphism. So we have 0 NG( f ) = NG( f ). An isomorphism i : ޑ5(G, X) ¯ → ޑ5(G, X × I ) is induced by the 0∗ φ,y 8,i0(y) inclusion i0, and analogously i1 induces an isomorphism. These isomorphisms do not change the number of essential classes. We have chG(X, f )(λG( f )) = −1 0 0 G G 0 (i0∗) i1∗ chG(X, f )(λG( f )), so N ( f ) = N ( f ). 

4. Lower bound property

The equivariant Nielsen invariants give a lower bound for the number of fixed point orbits on X H (x)\ X >H (x) and on X H (x), for maps lying in the G-homotopy class of f . Under mild hypotheses, this is even a sharp lower bound. 248 JULIA WEBER

Definition 4.1. Let G be a discrete group, let X be a finite proper G-CW-complex, and let f : X → X be a G-equivariant map. For every x¯ ∈ Is 5(G, X), with x : G/H → X, we set  MG( f )x¯ := min # fixed point orbits of ϕH (x) ϕ 'G f , G  H M ( f )x¯ := min # fixed point orbits of ϕ (x) ϕ 'G f .

H When speaking of fixed point orbits of f (x), we can either look at the WHx - H (H) orbits WHx · z ⊆ X (x) or at the G-orbits G · z ⊆ X (x), for a fixed point z in X H (x). These two notions are of course equivalent. We now proceed to show the first important property of the equivariant Nielsen invariants, the lower bound property. Proposition 4.2. For every x¯ ∈ Is 5(G, X) we have

G G NG( f )x¯ ≤ MG( f )x¯ , N ( f )x¯ ≤ M ( f )x¯ . H Proof. (1) If α ∈ π1(X (x), x)φ00 is an essential class of νG( f )x¯ , there has to be at least one fixed point orbit in X H (x) \ X >H (x) that corresponds to α and that >H cannot be moved into X (x). So, for any ϕ 'G f , the restriction ϕH must have H >H at least NG( f )x¯ fixed point orbits in X (x) \ X (x). We arrive at NG( f )x¯ ≤ {#fixed point orbits of ϕH } for all ϕ 'G f , so NG( f )x¯ ≤ MG( f )x¯ . H G (2) Let x¯ ∈ Is 5(G, X). Suppose that ϕ 'G f such that ϕ (x) has M ( f )x¯ H S K G fixed point orbits in X (x). Let Ꮿ ⊆ x¯≤y ¯ π1(X (y), y)φ0 such that N (ϕ)x¯ = G H N ( f )x¯ = #Ꮿ. If there were less than #Ꮿ fixed point orbits in X (x), there would be less that #Ꮿ essential classes and we could have chosen a smaller Ꮿ. So there are H at least #Ꮿ essential classes, and thus ϕ (x) has at least #Ꮿ fixed point orbits.  To prove the sharpness of this lower bound, we need certain hypotheses, which are usually introduced when dealing with these problems. Such conditions were first used in [Fadell and Wong 1988]. Some authors treat slightly weakened as- sumptions [Ferrario 1999; Ferrario 2003; Jezierski 1995; Wilczynski´ 1984]. We do not weaken the standard gap hypotheses in the context of functorial equivariant Lefschetz invariants since the standard gap hypotheses are not homotopy invariant. So an analog of Theorem 6.3 would not hold. Definition 4.3. Let G be a discrete group and let X be a cocompact smooth G- manifold. We say that X satisfies the standard gap hypotheses if for each x¯ ∈ Is 5(G, X), with x : G/H → X, the inequalities dim X H (x) ≥ 3 and dim X H (x)− dim X >H (x) ≥ 2 hold. Under these hypotheses, we can use an equivariant analog of Wecken’s classical method [1941] to coalesce fixed points. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 249

Lemma 4.4. Let G be a discrete group and let X be a cocompact proper smooth G- manifold satisfying the standard gap hypotheses. Let f : X → X be a G-equivariant map. Let ᏻ1 = Gx1 and ᏻ2 = Gx2 be two distinct isolated G-fixed point orbits, where x1 : G/H → X and x2 : G/K → X with x1 ≤ x2. Suppose that there are paths (σ1, [t1]) ∈ Mor(x, x1) and (σ2, [t2]) ∈ Mor(x, x2) for an x¯ ∈ 5(G, X), with : → [ ] ∗ = = [ ] ∗ x G/H X, such that (σ1, t1 ) 1x1 α (σ2, t2 ) 1x2 , i.e., that the fixed point H orbits induce the same α ∈ π1(X (x), x)φ0 . Then there exists a G-homotopy { ft } >(H) relative to X such that f0 = f and Fix f1 = Fix f0 − Gᏻ1.

Proof. Suppose first that x1 < x2. Then Mor(x1, x2) 6= ∅. By replacing x1 and x2 with other points in the orbit if necessary, we can suppose that there exists a H morphism (τ, [v]) ∈ Mor(x1, x2), where v is a path in X (x) with v1 = x1 and H v0 = x2 ◦ τ and τ : G/H → G/K is a projection. We know that v ' f ◦ v (relative endpoints). (This is an equivalent characterization of x1 and x2 belonging H >H to the same fixed point class [Jiang 1983, I.1.10].) Since x1 ∈ X (x)\ X (x) and >H H >H x2 ∈ X (x) and dim X (x) − dim X (x) ≥ 2, we may assume that v can be H >H chosen such that v((0, 1]) ⊆ X (x) \ X (x). We coalesce x1 and x2 along v as in [Wong 1991b, 1.1] and [Schirmer 1986, 6.1]. We can do this by only changing f in a (cone-shaped) neighborhood U(v) of v. Because of the proper action of G on X and the free action of WH on X H \ X >H , this neighborhood U(v) can be chosen such that in X H \ X >H it does not intersect its g-translates for g 6∈ H ≤ G. Taking the G-translates of U(v), we move ᏻ1 to ᏻ2 along the paths Gv in GU(v), not changing the map f outside GU(v). Now suppose x1 = x2. In this case, the result follows from [Wong 1991a, 5.4], H >H since X (x) \ X (x) is a free and proper WHx -space, where again the proper action of G on X ensures that we can find a neighborhood of a path from x1 to x2 such that the G/H-translates do not intersect.  From Lemma 4.4, we can conclude the sharpness of the lower bound given by the equivariant Nielsen invariants. Theorem 4.5. Let G be a discrete group. Let X be a cocompact proper smooth G- manifold satisfying the standard gap hypotheses. Let f : X → X be a G-equivariant endomorphism. Then G G MG( f )x¯ = NG( f )x¯ , M ( f )x¯ = N ( f )x¯ for all x¯ ∈ Is 5(G, X). Proof. (1) Since X is a cocompact smooth G-manifold, there is a G-map f 0 which is G-homotopic to f and which has only finitely many fixed point orbits. We apply Lemma 4.4 to f 0 to coalesce fixed point orbits in X H (x)\X >H (x) with others of the H >H same class α ∈ ޚπ1(X (x), x)φ0 . We move them into X (x) whenever possible. (We might need to create a fixed point orbit in the inessential fixed point class 250 JULIA WEBER beforehand; see [Wong 1991b, 1.1].) We remove the inessential fixed point orbits. We arrive at a map h 'G f such that NG( f )x¯ = #{fixed point orbits of h H (x)} ≥ MG( f )x¯ . Using Proposition 4.2, we obtain equality. (2) Since X is a cocompact smooth G-manifold, there is a map f 0 which is G- homotopic to f and which only has finitely many fixed point orbits. We have a partial ordering on the y¯ ≥x ¯ given by y¯ ≥z ¯ ⇔ Mor(z, y) 6= ∅. We apply Lemma 4.4 to f 0 to coalesce fixed point orbits of the same class, starting from the top. Note that when we remove fixed point orbits, we can only move them up in this partial ordering. That is why the definition has to be so complicated. We remove the inessential fixed point orbits. We are left with one fixed point orbit for every essential class. G We now look at a class Ꮿ such that N ( f )x¯ = #Ꮿ, and we coalesce the es- sential fixed point orbits with the corresponding classes appearing in Ꮿ. (If the corresponding class in Ꮿ is inessential, we might need to create a fixed point orbit in this inessential fixed point class beforehand.) We obtain a map h 'G f which has exactly #Ꮿ fixed point orbits. Hence G H G N ( f )x¯ = #{fixed point orbits of h (x)} ≥ M ( f )x¯ .

Using Proposition 4.2, we obtain equality. 

In general, it is not possible to find a map h 'G f realizing all minima simulta- neously. As an example, one can take G = ޚ/2 acting on X = S4 as an involution ޚ/2 3 4 so that X = S . One obtains MG(idS4 )x¯ = 0 for all x¯ ∈ Is 5(ޚ/2, S ), but the minimal number of fixed points in the G-homotopy class of the identity idS4 is equal to 1 [Wong 1993, Remark 3.4]. In this example, the standard gap hypotheses are not satisfied. Other examples where the standard gap hypotheses do not hold and where the converse of the equivariant Lefschetz theorem is false are given in [Ferrario 1999, Section 5].

5. The G-Jiang condition

In the nonequivariant case, we know that the generalized Lefschetz invariant is the right element when looking for a precise count of fixed points. We read off the Nielsen numbers from this invariant. In general, the Lefschetz number contains too little information. But under certain conditions, we can conclude facts about the Nielsen numbers from the Lefschetz numbers directly, and thus obtain a converse of the Lefschetz fixed point theorem. These conditions are called Jiang conditions. See [Jiang 1983, Definition II.4.1], where one can find a thorough treatment, and [Brown 1971, Chapter VII]. The Jiang group is a subgroup of π1(X, f (x)) [Jiang 1983, Definition II.3.5]. We generalize its definition to the equivariant case. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 251

Definition 5.1. Let G be a discrete group, let X be a finite proper G-CW-complex, and let f : X → X be a G-equivariant endomorphism. Then a G-equivariant self- H homotopy h : f 'G f of f determines a path h(x, −) ∈ π1(X (x), f (x)) for every x¯ ∈ Is 5(G, X), with x : G/H → X. Define the G-Jiang group of (X, f ) to be   X   JG(X, f ) := h(x, −) h : f 'G f G-equivariant self-homotopy

x¯∈5(G,X) M H
≤ π1 X (x), f (x) , x¯∈5(G,X) and define the G-Jiang group of X to be   X   JG(X) := h(x, −) h : id 'G id G-equivariant self-homotopy

x¯∈5(G,X) M H
≤ π1 X (x), x . x¯∈5(G,X) In the nonequivariant case, we know that the Jiang group J(X, f, x) is a sub-

group of the centralizer of π1( f, x) π1(X, x) in π1 X, f (x) . In particular,
J(X) ≤ Z π1(X, x) ,
where Z π1(X, x) denotes the center of π1(X, x) [Jiang 1983, Lemma II.3.7].

Furthermore, the isomorphism ( f ◦w)∗ : π1 X, f (x1) → π1 X, f (x0) induced by a path w from x0 to x1 induces an isomorphism ( f ◦w)∗ : J(X, f, x1)→ J(X, f, x0) which does not depend on the choice of w. So the definition does not depend on the choice of the basepoint [Jiang 1983, Lemma II.3.9]. It is also known that J(X) ≤ J(X, f ) ≤ π1(X) for all f [Jiang 1983, Lemma II.3.8]. This leads to the consideration of spaces with J(X) = π1(X) in the definition of a Jiang space. All these lemmata also make sense in the equivariant case. Thus we make the following definition. Definition 5.2. Let G be a discrete group and let X be a cocompact G-CW- complex. Then X is called a G-Jiang space if for all x¯ ∈ Is 5(G, X) we have H
JG(X)x¯ = π1 X (x), x . H H The group JG(X, f ) acts on 3G(X, f ) as follows: If X ( f (x)) = X (x) and H H H X ( f (x)) = X (x), then JG(X, f )x¯ acts on ޚπ1(X (x), x). The element u = −1 [h(x, −)] ∈ JG(X, f )x¯ acts as composition with [wuw ], where v = (id, [w]) ∈ Mor( f (x), x). H H
Since JG(X, f )x¯ is contained in the centralizer of π1( f (x), x) π1(X (x), x) H
H in π1 X (x), f (x) , this action induces an action on ޚπ1(X (x), x)φ0 by compo- sition, whence on 3G(X, f )x¯ . Thus JG(X, f ) acts on 3G(X, f ), and by invariance 252 JULIA WEBER of λG( f ) under homotopy equivalence, we see that

JG (X, f ) λG( f ) ∈ 3G(X, f ) .

Examples of G-Jiang spaces can be obtained from Jiang spaces. It is known [Jiang 1983, Theorem II.3.11] that the class of Jiang spaces is closed under homo- topy equivalence and the topological product operation and contains

• simply connected spaces, • generalized lens spaces, • H-spaces,

• homogeneous spaces of the form A/A0 where A is a topological group and A0 is a subgroup which is a connected compact Lie group. Hence we obtain many examples of G-Jiang spaces using the following propo- sition, analogous to [Wong 1993, Proposition 4.9].

Proposition 5.3. Let G be a discrete group, and let X be a free cocompact con- nected proper G-space. If X/G is a Jiang space, then X is a G-Jiang space.

Proof. Since X is connected and free, the set Is 5(G, X) consists of one element. p Let x be a basepoint of X. We need to check that JG(X)x¯ = π1(X, x). Let X −→ X/G be the projection. The Jiang subgroup of X/G is given by n o :=  −  : ' J(X/G) h(p(x), ) h idX/G idX/G self-homotopy
≤ π1 X/G, p(x) .

p Let α ∈ π1(X, x). Since X −→ X/G is a discrete cover, eX = X]/G. There is a map p# : π1(X, x) → π1(X/G, p(x)) induced by the projection. Since X/G is a Jiang space, J(X/G) = π1(X/G, p(x)), so there is a homotopy h : idX/G ' idX/G such that p#(α) = [h(p(x), −)]. Because of the free and proper action of G on X, this 0 homotopy h can be lifted to a G-equivariant homotopy h : idX 'G idX such that 0 α = [h (x, −)]. Thereby α ∈ JG(X). 

6. The converse of the equivariant Lefschetz Fixed Point Theorem

One can derive equivariant analogs of statements about Nielsen numbers found in [Jiang 1983], generalizing results from [Wong 1993] to infinite discrete groups. In particular, if X is a G-Jiang space, the converse of the equivariant Lefschetz fixed point theorem holds. The next theorem can be compared with [Jiang 1983, Theorem II.4.1]. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 253

Theorem 6.1. Let G be a discrete group, and let X be a finite proper G-CW- complex which is a G-Jiang space. Then for any G-map f : X → X and x¯ ∈ Is 5(G, X) with x : G/H → X we have:

LG( f )x¯ = 0 H⇒ λG( f )x¯ = 0 and NG( f )x¯ = 0,  H LG( f )x¯ 6= 0 H⇒ λG( f )x¯ 6= 0 and NG( f )x¯ = # π1(X (x), x)φ00 .

Here LG( f ) is the equivariant Lefschetz class [Lück and Rosenberg 2003a, Defi- nition 3.6], the equivariant analog of the Lefschetz number.

Proof. Since X is a G-Jiang space, the G-Jiang group JG(X) acts transitively on H π1(X (x), x) for all x¯ ∈ Is 5(G, X). This implies that X X λG( f )x¯ = nα · α = n · α α α

 H for some n ∈ ޚ. This leads to LG( f )x¯ = n·# π1(X (x), x)φ0 by the augmentation map. We see that

LG( f )x¯ = 0 H⇒ n = 0

H⇒ λG( f )x¯ = 0

H⇒ νG( f )x¯ = 0

H⇒ NG( f )x¯ = 0,

LG( f )x¯ 6= 0 H⇒ n 6= 0 H H⇒ λG( f )x¯,α 6= 0 for all α ∈ ޚ(π1(X (x), x))φ0 H H⇒ νG( f )x¯,α 6= 0 for all α ∈ ޚ(π1(X (x), x))φ00  H H⇒ NG( f )x¯ = # π1(X (x), x)φ00 . 

The proof of Theorem 6.1 already works if JG(X, f ) acts transitively on every summand of 3G(X, f ). We could have called X a G-Jiang space if the condition that JG(X, f ) acts transitively on every summand of 3G(X, f ) is satisfied. But H this condition is less tractable. It is implied by JG(X, f )x¯ = π1(X (x), f (x)) for H all x¯, which is implied by JG(X)x¯ = π1(X (x), x) for all x¯. We now show that f is G-homotopic to a fixed point free G-map if the gener- alized equivariant Lefschetz invariant λG( f ) is zero.

Theorem 6.2. Let G be a discrete group. Let X be a cocompact proper smooth G- manifold satisfying the standard gap hypotheses. Let f : X → X be a G-equivariant endomorphism. If λG( f ) = 0, then f is G-homotopic to a fixed point free G-map. 254 JULIA WEBER

G Proof. If λG( f )=0, then chG(X, f )(λG( f ))=0, and therefore we have N ( f )x¯ = G G 0 for all x¯ ∈ Is 5(G, X). We know from Theorem 4.5 that N ( f )x¯ = M ( f )x¯ = H min{# fixed point orbits of ϕ (x) | ϕ 'G f }. In particular, for x : G/{1} → X we {1} obtain a map ϕ such that ϕ (x) is fixed point free and ϕ 'G f . Thus we obtain our result on every connected component of X, and combining these we arrive at a map h 'G f which is fixed point free.  These two theorems, Theorem 6.1 and Theorem 6.2, combine to give the main theorem of this paper, the converse of the equivariant Lefschetz fixed point theorem. Theorem 6.3. Let G be a discrete group. Let X be a cocompact proper smooth G-manifold satisfying the standard gap hypotheses which is a G-Jiang space. Let f : X → X be a G-equivariant endomorphism. Then the following holds:

If LG( f ) = 0, then f is G-homotopic to a fixed point free G-map.

Proof. We know that LG( f ) = 0 means that LG( f )x¯ = 0 for all x¯ ∈ Is 5(G, X). Since X is a G-Jiang space, by Theorem 6.1 this implies that λG( f )x¯ = 0 for all x¯ ∈ Is 5(G, X), so we have λG( f ) = 0. We apply Theorem 6.2 to arrive at the desired result.  Remark 6.4. As another corollary of Theorem 6.2, we obtain: If G is a discrete group and X is a cocompact proper smooth G-manifold satisfying the standard G gap hypotheses, then χ (X) = 0 implies that the identity idX is G-homotopic to a fixed point free G-map. This was already stated in [Luck¨ and Rosenberg 2003a, Remark 6.8]. Here χ G(X) is the universal equivariant Euler characteristic G of X [Luck¨ and Rosenberg 2003a, Definition 6.1] defined by χ (X)x¯ = χ WHx \ H >H
G X (x), WHx \ X (x) ∈ ޚ, we have χ (X) = LG(idX ). We calculate that

X p X
λG(idX )x¯ = (−1) incφ id , e X^H (x) p≥0 Aut(x)·e∈ Aut(x)\Ip(X^H (x),X^>H (x)) = \ H \ >H
· ∈ H
χ WHx X (x), WHx X (x) 1 ޚπ1 X (x), x φ0 .

G So we have χ (X) = 0 if and only if λG(idX ) = 0, and with Theorem 6.2 we conclude that there is an endomorphism G-homotopic to the identity which is fixed point free.

References

[Brown 1971] R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman, Glenview, IL, 1971. MR 44 #1023 [tom Dieck 1979] T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics 766, Springer, Berlin, 1979. MR 82c:57025 Zbl 0445.57023 EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 255

[tom Dieck 1987] T. tom Dieck, Transformation groups, Studies in Mathematics 8, de Gruyter, Berlin, 1987. MR 89c:57048 Zbl 0611.57002 [Fadell and Wong 1988] E. Fadell and P. Wong, “On deforming G-maps to be fixed point free”, Pacific J. Math. 132:2 (1988), 277–281. MR 89d:55003 [Ferrario 1999] D. L. Ferrario, “A fixed point index for equivariant maps”, Topol. Methods Nonlinear Anal. 13:2 (1999), 313–340. MR 2001b:55004 Zbl 0959.55004 [Ferrario 2003] D. L. Ferrario, “A Möbius inversion formula for generalized Lefschetz numbers”, Osaka J. Math. 40:2 (2003), 345–363. MR 2004d:55001 [Ferrario 2005] D. L. Ferrario, “A note on equivariant fixed point theory”, pp. 287–300 in Handbook of topological fixed point theory, edited by F. Brown et al., Springer, Dordrecht, 2005. MR 2006e: 55003 Zbl 1082.55001 [Hart 1999] E. L. Hart, “The Reidemeister trace and the calculation of the Nielsen number”, pp. 151–157 in Nielsen theory and Reidemeister torsion (Warsaw, 1996), edited by J. Jezierski, Banach Center Publ. 49, Polish Acad. Sci., Warsaw, 1999. MR 1734729 Zbl 0943.55001 [Jezierski 1995] J. Jezierski, “A modification of the relative Nielsen number of H. Schirmer”, Topol- ogy Appl. 62:1 (1995), 45–63. MR 95m:55002 [Jiang 1983] B. J. Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 14, American Mathematical Society, Providence, R.I., 1983. MR 84f:55002 Zbl 0512.55003 [Lück 1989] W. Lück, Transformation groups and algebraic K -theory, Lecture Notes in Mathemat- ics 1408, Springer, Berlin, 1989. MR 91g:57036 Zbl 0679.57022 [Lück and Rosenberg 2003a] W. Lück and J. Rosenberg, “The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds”, pp. 322–361 in High-dimensional manifold topol- ogy (Trieste, 2001), edited by F. T. Farrell and W. Lück, World Scientific, River Edge, NJ, 2003. MR 2005b:57064 Zbl 1046.57026 [Lück and Rosenberg 2003b] W. Lück and J. Rosenberg, “Equivariant Euler characteristics and K - homology Euler classes for proper cocompact G-manifolds”, Geom. Topol. 7 (2003), 569–613. MR 2004k:19005 Zbl 1034.19003 [Nielsen 1920] J. Nielsen, “Über die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ring- flächen”, Math. Ann. 82:1-2 (1920), 83–93. MR 1511973 JFM 47.0527.03 [Reidemeister 1936] K. Reidemeister, “Automorphismen von Homotopiekettenringen”, Math. Ann. 112:1 (1936), 586–593. MR 1513064 JFM 62.0659.05 [Schirmer 1986] H. Schirmer, “A relative Nielsen number”, Pacific J. Math. 122:2 (1986), 459–473. MR 87e:55002 Zbl 0553.55001 [Weber 2005] J. Weber, “The universal functorial equivariant Lefschetz invariant”, K -Theory 36:1-2 (2005), 169–207. MR 2274162 Zbl 05125362 [Wecken 1941] F. Wecken, “Fixpunktklassen, II: Homotopieinvarianten der Fixpunkttheorie”, Math. Ann. 118 (1941), 216–234. MR 5,275a JFM 67.1094.01 [Wilczynski´ 1984] D. Wilczynski,´ “Fixed point free equivariant homotopy classes”, Fund. Math. 123:1 (1984), 47–60. MR 86i:57046 Zbl 0548.55002 [Wong 1991a] P. Wong, “Equivariant Nielsen fixed point theory for G-maps”, Pacific J. Math. 150:1 (1991), 179–200. MR 92i:55005 Zbl 0691.55004 [Wong 1991b] P. Wong, “On the location of fixed points of G-deformations”, Topology Appl. 39:2 (1991), 159–165. MR 92c:55004 Zbl 0722.55002 [Wong 1993] P. Wong, “Equivariant Nielsen numbers”, Pacific J. Math. 159:1 (1993), 153–175. MR 94e:55006 Zbl 0739.55001 256 JULIA WEBER

Received November 16, 2005. Revised June 21, 2006.

JULIA WEBER MAX-PLANCK-INSTITUTFUR¨ MATHEMATIK VIVATSGASSE 7 D-53111 BONN GERMANY [email protected]